Nash Equilibrium Seeking Algorithm Research Articles

Distributed Nash equilibrium seeking for quadratic games in discrete-time systems with bounded control inputs

This paper considers the distributed Nash equilibrium seeking problem for quadratic games in discrete-time systems with bounded control inputs. First, a saturation gradient algorithm is proposed to seek the Nash equilibrium without considering the limitations of communication. Then the case that players can only communicate with their neighbors is considered, a distributed Nash equilibrium seeking algorithm is designed where a consensus protocol is adapted for information sharing. In the proposed distributed algorithm, each player has an estimate on others’ actions and the consensus of players’ estimates is achieved. By Lyapunov stability theory for discrete-time systems, it is shown that the Nash equilibrium of the game is globally asymptotically stable under certain conditions. Moreover, distributed Nash equilibrium seeking problem in hybrid systems with bounded control inputs is solved. Finally, two numerical examples are presented to verify the effectiveness of the proposed algorithms.

Resilient Discrete‐Time Quantization Communication for Distributed Nash Equilibrium Seeking Under DoS Attacks

ABSTRACTThis paper studies the design of resilient distributed Nash equilibrium (NE) algorithm for second‐order multi‐agent systems in the presence of Denial‐of‐Service (DoS) attacks, which prevent information transmission in the network with limited bandwidth and transmission rate. Resilient communication schemes, integrating periodic/event‐triggered schemes and a dynamic quantized transmission method, are designed in a distributed NE seeking algorithm to ensure the convergence of agents' strategies to the NE of the game. For the designed two resilient communication schemes, the sufficient conditions for the convergence of the algorithm are established, and the tradeoff between the sampling period and the frequency and duration of DoS attacks is analyzed. Cournot competition with distributed energy resources is used as an example to verify the proposed methods.

Multi-Cluster Aggregative Games: A Linearly Convergent Nash Equilibrium Seeking Algorithm and Its Applications in Energy Management

Multi-Cluster Aggregative Games: A Linearly Convergent Nash Equilibrium Seeking Algorithm and Its Applications in Energy Management

Distributed Nash Equilibrium Seeking Dynamics With Discrete Communication.

In this brief, we aim to provide a distributed Nash equilibrium seeking algorithm in continuous time with discrete communications. A group of agents are considered playing a continuous-kernel noncooperative game over a network. The agents need to seek the Nash equilibrium when each player cannot get the overall action profiles in real time, rather are only able to get information from its networked neighbors. Meanwhile, a continuous-time dynamics is discussed for the players to update their variables, but the communications over the network are only assumed to allow at discrete-time instants, since continuous-time communications are prohibitive and cumbersome in practice. First, the periodic communication is considered at a fixed interval, and the solvability of Nash equilibrium seeking is shown with discrete communications. Then, an event-trigger communication scheme is proposed to further reduce the communication rounds. Nevertheless, the event-trigger communication scheme requires each player continuously monitoring its local states. To alleviate the monitoring burden, a periodic event detection mechanism is further developed. The exponential convergence of the dynamics with the three discrete communication schemes is proven. Finally, the comparative simulation studies are designed to illustrate the algorithm performance with different communication schemes and parameter settings.

An Algorithm for Resilient Nash Equilibrium Seeking in the Partial Information Setting

Current research in distributed Nash equilibrium (NE) seeking in the partial information setting assumes that information is exchanged between agents that are “truthful”. However, in general noncooperative games agents may consider sending misinformation to neighboring agents with the goal of further reducing their cost. Additionally, communication networks are vulnerable to attacks from agents outside the game as well as communication failures. In this paper, we propose a distributed NE seeking algorithm that is robust against adversarial agents that transmit noise, random signals, constant singles, deceitful messages, as well as being resilient to external factors such as dropped communication, jammed signals, and man in the middle attacks. The core issue that makes the problem challenging is that agents have no means of verifying if the information they receive is correct, i.e. there is no “ground truth”. To address this problem, we use an observation graph, that gives truthful action information, in conjunction with a communication graph, that gives (potentially incorrect) information. By filtering information obtained from these two graphs, we show that our algorithm is resilient against adversarial agents and converges to the Nash equilibrium.

Distributed generalized Nash equilibrium seeking: A backward-reflected-forward-backward-based algorithm

This paper studies non-cooperative games, in which each player’s local objective function depends on its own decisions as well as those of other players. In these games, each player’s local objective function is differentiable, its decision is constrained by a local feasibility constraint, and the decisions of all players are coupled with an equality constraint. By analyzing the variational problem of the games, a distributed generalized Nash equilibrium seeking algorithm with fixed step sizes is developed based on the backward-reflected-forward–backward splitting. Each player performs a backward step followed by a forward–backward step, and the pseudo-gradient is evaluated at a reflection term. Moreover, the convergence of the proposed distributed algorithm is analyzed under standard assumptions using the operator theory and the convex analysis theory. Finally, the simulation results verify the effectiveness of the algorithm and the correctness of the theoretical analysis.

Distributed fixed-time Nash equilibrium seeking algorithm for multiple ASVs: A hybrid event-triggered approach

In this article, the fixed-time Nash equilibrium (FTNE) seeking problem of multiple autonomous surface vehicles (ASVs) is considered. As individuals fight for an allocation of specific shared but restricted resources, two novel FTNE seeking algorithms are designed for this topic through an error decomposition method. First, a model-based control algorithm with only one convergence time function in a single-layer structure is designed to achieve Nash equilibrium within a fixed time. Then, a FTNE algorithm with a three-layer structure is presented for multiple ASVs subject to system uncertainties. In particular, this paper proposes a hybrid fixed-time event-triggered approach for reducing controller update frequencies and communication frequencies of two algorithms simultaneously, which is capable of decreasing resource consumption in a constrained resource environment. Furthermore, rigorous sufficient criteria are established for fixed-time convergence through Lyapunov stability analysis, and the upper bound on settling time without the requirement of initial conditions is derived. Finally, simulation examples are given to demonstrate the viability of two algorithms.

Hybrid Nash Equilibrium Seeking Under Partial-Decision Information: An Adaptive Dynamic Event-Triggered Approach

This paper is concerned with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hybrid</i> Nash equilibrium (NE) seeking problem over a network in a partial-decision information scenario. Each agent has access to both its own cost function and local decision information of its neighbors. First, an adaptive gradient-based algorithm is constructed in a fully distributed manner with the guaranteed convergence to the NE, where the network communication is required. Second, in order to save communication cost, a novel event-triggered scheme, namely, edge-based adaptive dynamic event-triggered (E-ADET) scheme, is proposed with on-line-tuned triggering parameter and threshold, and such a scheme is proven to be fully distributed and free of Zeno behavior. Then, a hybrid NE seeking algorithm, which is also fully distributed, is constructed under the E-ADET scheme. By means of the Lipschitz continuity and the strong monotonicity of the pseudo-gradient mapping, we show the convergence of the proposed algorithms to the NE. Compared with the existing distributed algorithms, our algorithms remove the requirement on global information, thereby exhibiting the merits of both flexibility and scalability. Finally, two examples are provided to validate the proposed NE seeking methods.

Agents Attraction Competition in an Extended Friedkin-Johnsen Social Network

Unlike many complex social networks investigated in the existing literatures concerning about the communication, consensus, or cooperative behaviors, this paper is devoted to an agents attraction competition problem in an extended Friedkin-Johnsen network. In this problem, two strategic agents compete to attract the nonstrategic agents to adopt opinions as closer to theirs as possible, by connecting with these nonstrategic agents and allocating their strengths on these connections. Therefore a 2-person zero-sum competitive game is characterized. We develop mathematical tools to provide some useful properties of the game, including the partial derivatives and positive definiteness of the payoff functions, based on which the Nash equilibrium of the game is analyzed. We rigorously examine that each player will allocate their strengths according to the susceptibilities, the intrinsic opinions and the costs of the nonstrategic agents, and the weighted column-sums of the row-stochastic matrix which describes the interpersonal influences of the network, if the network satisfies a condition. Otherwise, a Nash equilibrium seeking algorithm is proposed with simplex constraints, by which each strategic agent can asymptotically learn its optimal strategy. Finally, numerical examples are presented to illustrate the validity of the obtained results.

Byzantine Agents’ Detection in Distributed Nash Equilibrium Seeking Algorithms Using an Adaptive Event-Triggered Scheme

In this article, a resilient, adaptive event-triggered distributed Nash equilibrium seeking algorithm is investigated for a class of multimicrogrid systems. Since the strategy of each microgrid affects others’ costs through total generation cost, such interaction among microgrids is modeled as a noncooperative game. To reduce unnecessary signal transmission among the aggregators, an adaptive algorithm, where threshold parameters are adjusted based on the behavior of the estimated signals in the last two iterations, is proposed. It is observed that when false data is injected into an agent, its estimated signals begin to diverge from the equilibrium point, and as a result, the threshold value becomes increasing. Furthermore, the misbehaving agents are identified and isolated using a reputation-based mechanism and are allowed to rejoin the network when they behave normally again. To deal with arbitrary behavior of users, receding horizon scheme of predictive control is used. Finally, a set of simulation results is presented to evaluate the effectiveness of the suggested algorithm.

Fixed-time consensus-based distributed Nash equilibrium seeking for noncooperative game with second-order players

This paper aims to study the Nash equilibrium (NE) seeking algorithm of noncooperative game with second-order system under strongly connected digraph. In the algorithm, velocity information is not directly accessible to players, in order to avoid the need for additional velocity observers. Meanwhile, for position information, the position estimation of all the other players under local communication is completed in fixed time. To address the aforementioned concern, a NE seeking algorithm with a two-time-scale structure is designed. (1) For reduced system, a second-order velocity estimation model based on state observer is introduced to avoid direct acquisition of velocity information; (2) For fast compensation mechanism, each player uses leader-following consensus protocol to estimate the position of all the other players. The fixed-time consensus protocol is utilized to ensure that the position is estimated in fixed time. Then, aiming to reduce the communication costs and communication burden among players, the event-triggered mechanism is introduced into the above algorithm, and Zeno behavior is excluded. Lastly, a self-organized network problem involving autonomous vehicles is presented to validate the efficacy of the proposed theoretical results.

Distributed Robust Nash Equilibrium Seeking for Mixed-Order Games by a Neural-Network-Based Approach

In practical applications, decision makers with heterogeneous dynamics may be engaged in the same decision-making process. This motivates us to study distributed Nash equilibrium seeking for games in which players are mixed-order (first-and second-order) integrators influenced by unknown dynamics and external disturbances in this article. To solve this problem, we employ an adaptive neural network to manage unknown dynamics and disturbances, based on which a distributed Nash equilibrium seeking algorithm is developed by further adapting concepts from gradient-based optimization and multiagent consensus. By constructing appropriate Lyapunov functions, we analytically prove the convergence of the reported method. Theoretical investigations suggest that players’ actions would be steered to an arbitrarily small neighborhood of the Nash equilibrium, which is also testified by simulations.

A distributed coding-decoding-based Nash equilibrium seeking algorithm over directed communication network

A distributed coding-decoding-based Nash equilibrium seeking algorithm over directed communication network

Distributed Nash Equilibrium Seeking for Aggregative Games With Quantization Constraints

The problem of seeking Nash equilibrium (NE) based on aggregative games under quantization constraints is full of challenges. Although the NE seeking algorithm in continuous-time systems has been studied, this problem in discrete-time systems still needs to be solved urgently. To address this problem, three distributed algorithms are first proposed under three quantization cases, adaptive, random, and time-varying quantizations, based on doubly stochastic communication topology networks. Then, the actions of players would eventually converge to NE under the conditions of vanishing step size and strong monotonicity are proved. Moreover, the convergence rate of the three quantization cases are analyzed, respectively. Finally, numerical experiments are implemented on plug-in hybrid electric vehicles (PHEVs) to validate the effectiveness of the proposed distributed algorithms. Comparing the convergence rates of the three proposed algorithms, the convergence effect of the adaptive quantization is better than that of the other two quantization cases.

Distributed Computation of Stochastic GNE With Partial Information: An Augmented Best-Response Approach

In this paper, we focus on the stochastic generalized Nash equilibrium problem (SGNEP) which is an important and widely-used model in many different fields. In this model, subject to certain global resource constraints, a set of self-interested players aim to optimize their local objectives that depend on their own decisions and the decisions of others and are influenced by some random factors. We propose a distributed stochastic generalized Nash equilibrium seeking algorithm in a partial-decision information setting based on the Douglas-Rachford operator splitting scheme, which relaxes assumptions in the existing literature. The proposed algorithm updates players' local decisions through augmented best-response schemes and subsequent projections onto the local feasible sets, which comprise most of the computational workload. The projected stochastic subgradient method is applied to provide approximate solutions to the augmented best-response subproblems for each player. The Robbins-Siegmund theorem is leveraged to establish the main convergence results to a true Nash equilibrium using the proposed inexact solver. Finally, we illustrate the validity of the proposed algorithm via two numerical examples, i.e., a stochastic Nash-Cournot distribution game and a multi-product assembly problem with the two-stage model.

A Fully Distributed Nash Equilibrium Seeking Algorithm for N-Coalition Games of Euler–Lagrange Players

This article investigates the N-coalition games of Euler–Lagrange (EL) systems. In the games, every coalition consists of a group of players, whose dynamics are formulated as EL equations. The objective of each coalition is to minimize its cost function, which is the sum of local cost functions of the players in the corresponding coalition. That is, the players within the coalition collaboratively minimize their coalition’s cost function. The cost function of each player is only available to itself and depends on not only its decision, but also the aggregate of all decisions. Based on gradient descent, consensus technique, and state feedback, a fully distributed Nash equilibrium seeking algorithm for N-coalition games is proposed. The exponential convergence of the algorithm is analyzed by the Lyapunov stability theorem. A numerical example in smart grids is given to illustrate the effectiveness of the proposed algorithm.

Generalized multi-cluster game under partial-decision information with applications to management of energy internet

The decision making and management of many engineering networks involves multiple parties with conflicting interests, while each party is constituted by multiple agents. Such problems can be cast as a multi-cluster game. Each cluster is treated as a self-interested player in a non-cooperative game where agents in the same cluster cooperate to optimize the cost function of the cluster. In a large-scale network, the information of agents in a cluster can not be available immediately to agents beyond this cluster, which raises challenges to the existing Nash equilibrium seeking algorithms. Hence, we consider a partial-decision information scenario in multi-cluster games. We reformulate the problem by finding zeros of the sum of preconditioned monotone operators by the primal-dual analysis and graph Laplacian matrix. Then a distributed generalized Nash equilibrium seeking algorithm is proposed without requiring full awareness of its opponent clusters’ decisions based on a forward-backward-forward method. With the algorithm, each agent estimates the strategies of all the other clusters by communicating with neighbors via an undirected network. We show that the derived operators can be monotone when the communication strength parameter is sufficiently large. We prove the algorithm convergence by providing a sufficient condition of the fixed point theory. We discuss its potential application in Energy Internet with numerical studies.

An adaptive generalized Nash equilibrium seeking algorithm under high-dimensional input dead-zone

In this paper, a novel adaptive generalized Nash equilibrium (GNE) seeking algorithm is designed, in order to address the non-cooperative game with private inequality constraints under high-dimensional input dead-zone. That is to say, the dead-zone dynamics may be thought of as a generic high-dimensional convex set, and the introduction of two methods distinguishes our works in seeking the GNE of non-cooperative games. On the one hand, a two-time-scale structure based on singular perturbation method is led into the design of GNE seeking algorithm, where the fast dynamics part rapidly eliminates the influence of input dead-zone, and the slow dynamics part drives the players’ action to the GNE. On the other hand, adaptive penalty method is utilized to ensure the player’s action enters the inequality constraints set without a prior estimation of centralized information for penalty parameters. The algorithm in this paper realizes complete distribution and parameter independence, making it easy to apply in practical programming. At last, several numerical examples regarding the electricity markets are employed to verify the effectiveness of the theoretical results.

Adaptively Distributed Nash Equilibrium Seeking of Noncooperative Games for Uncertain Heterogeneous Linear Multi-Agent Systems

This paper presents the design of adaptively distributed Nash Equilibrium (NE) seeking algorithms in noncooperative games for heterogeneous general linear multi-agent systems (MASs) under unknown unmodeled dynamics and bounded disturbances. Different from existing works that only consider single or multiple integrators, we aim to steer agents' outputs of MASs with nonidentical dynamics to the NE in a distributed way and not needing known information on the Lipschitz and monotone constants of pseudo-gradients as well as the algebraic connectivity of the graph. To overcome difficulties brought by heterogeneous dynamics and NE seeking requirements, we first present an adaptively distributed NE seeking algorithm that can tune on-line the edges of graphs to solve the studied problem. By leveraging monotone and matrix properties, the global asymptotic convergence to the NE is obtained. Moreover, this design is extended to develop another adaptively distributed NE seeking algorithm to tackle the impact of unknown dynamics and disturbances. Two exam -ples with numerical simulation results are provided to illustrate the effectiveness of the developed NE seeking algorithms.

Resilient Nash Equilibrium Seeking in Multiagent Games Under False Data Injection Attacks

This article proposes a resilient distributed Nash equilibrium (NE) seeking algorithm for noncooperative games with multiple double-integrator agents who suffer from false data injection (FDI) attacks. A malicious attacker injects false data into agents’ actuators and sensors so that agents’ strategies deviate from the NE of the game with the compromised control inputs and interactive information. First, the robustness of the seeking algorithm against the FDI attacks is analyzed. Then, to mitigate the effect of the attacks on agents’ strategies, the false data injected in the actuators and sensors are regarded as extended states which can be observed by extended state observers (ESOs). Thus, a resilient NE seeking algorithm is proposed based on ESOs. The resilient algorithm can drive the system to converge to the NE without requiring any information about the nature of the attacks. An explicit criterion is given to ensure the effectiveness of the designed algorithms. An example is given to illustrate the results.